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## Homework Statement

An example of a weak decay of a nucleus is (Z, A) → (Z +1, A) + τ

^{-}+ ν

_{τ}-bar where (Z,A) represents a nucleus with Z protons and A−Z neutrons.

(d) Suppose the gauge boson in the above process is produced at rest. Assuming the neutrino is massless, what must the 3-momenta of the τ

^{-}and neutrino be? [5 Marks]

(e) Now suppose that the neutrino is not massless. Express the mass of the gauge boson in terms of the lepton and neutrino masses and their 3-momenta. [3 Marks]

(f) Consider the case where the nucleus (Z,A) is at rest and the nucleus (Z +1,A) recoils with energy E along the X direction. The τ is observed to be moving along the Y direction (i.e. at right-angles to the nucleus’s recoil). What will the component of the 3-momenta of the neutrino be along the axis of the nucleus’s recoil? You should express your result in terms of E,Z,A,Mp and Mn. [7 Marks]

## Homework Equations

E before= E after

3-momenta before= 3-momenta after

4-momentum before= 4 momentum after

## The Attempt at a Solution

(d) Isolating the W_ bit of the process: W

^{-}= τ

^{-}+ ν

^{τ}-bar

so (M

_{W-},

__0__) = (E

_{τ_},

__P__) + (E

_{τ_}_{ντ},

__P__

_{ντ})And from 3-momentum conservation, the momenta of the neutrino and taon must be equal and opposite.

(My first issue is that i'm not sure that I can just isolate this part of the interaction, I just made an assumption from the Feynman diagram I drew). I also don't understand the significance of the neutrino being massless other than the |k

_{τ}| = E

_{ν(bar)τ}?

(e) Applying 4-momentum conservation: M

_{W}

^{2}= (E

_{τ}+ E

_{ν(bar)τ})

^{2}- (

__K__

_{τ}+

__K__

_{ν(bar)τ})

^{2}

But I can't seem to work this through to gain an answer in terms of lepton neutrino masses and their 3-momentum.

(f) I'm not sure how to even approach this part, I cant figure out whether I need to write the 4 momenta's using A-Z etc. Any kind of hint or guidance would be great so I can have a good stab.

Thanks in advance for any help, it's greatly appreciated!